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Chapter 7: Problem 67

Perform the indicated operation or operations. $$\frac{3 x^{2}+3 x-60}{2 x-8} \div\left(\frac{30 x^{2}}{x^{2}-7 x+10} \cdot \frac{x^{3}+3 x^{2}-10 x}{25 x^{3}}\right)$$

Short Answer

Expert verified
The solution to the given expression is \(\frac{3}{4x - 16}\).

Step by step solution

01

Step 1. Simplify the Fractions

First, factorize and simplify the given complex fractions: \[\frac{3 x^{2}+3 x-60}{2 x-8}\] becomes \[\frac{3(x^2+x-20)}{2(x-4)}\] and further simplifies to \[\frac{3x(x+5)}{2(x-4)}\] Similarly \[\frac{30 x^{2}}{x^{2}-7 x+10} \cdot \frac{x^{3}+3 x^{2}-10 x}{25 x^{3}}\] can be factorized and simplified to \[\frac{30x(x)}{(x-5)(x-2)} \cdot \frac{x(x^2+3x -10)}{25x(x^2)}\]which further simplifies to \[\frac{30x}{(x-5)(x-2)} \cdot \frac{x(x+5)(x-2)}{25x}\]
02

Step 2. Perform Division Operation

Remember that division is the same as multiplying by the reciprocal. So, the expression becomes\[\frac{3x(x+5)}{2(x-4)} \cdot \frac{(x-5)(x-2)}{30x} \cdot \frac{25x}{x(x+5)(x-2)}\]
03

Step 3. Simplify Expression

Now cancel out common factors from numerator and denominator to simplify the expression. Cancel x and (x+5) from the numerator and denominator and also simplify 30 and 25 to get \[\frac{3}{2(x-4)} * \frac{1}{2} = \frac{3}{4(x-4)}\]
04

Step 4. Final Simplification

Finally, for the most simplified form, distribute the fraction across the term in the denominator to get \[\frac{3}{4x - 16}\]

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Most popular questions from this chapter

Add or subtract as indicated. Simplify the result, if possible. $$\frac{y}{y^{2}+2 y+1}+\frac{4}{y^{2}+5 y+4}$$

Add or subtract as indicated. Simplify the result, if possible. Perform the indicated operation or operations. Simplify the result, if possible. $$\frac{x+6}{x^{2}-4}-\frac{x+3}{x+2}+\frac{x-3}{x-2}$$

Add or subtract as indicated. Simplify the result, if possible. $$7+\frac{1}{x-5}$$

Simplify: \(\left(3 x^{2}\right)\left(-4 x^{-10}\right) .\) (Section 5.7, Example 3)

A company that manufactures wheelchairs has fixed costs of \(\$ 500,000 .\) The average cost per wheelchair, \(C,\) for the company to manufacture \(x\) wheelchairs per month is modeled by the formula $$ C=\frac{400 x+500,000}{x} $$ Use this mathematical model to solve Exercises \(69-70\). How many wheelchairs per month can be produced at an average cost of \(\$ 405\) per wheelchair?
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